Here in Canada/US, a vest is basically the third piece of a three-piece suit, the thing that goes over a man's shirt but under his suit jacket. A sweater vest would be a man's knitted sweater, sleeveless and v-neck, to be worn over a collared shirt.

From what I gather, in the UK a "vest" is what we would call a tank top here? I did see the word "camisole" being used in the same way I would have used it, but the "vest" thing had me confused.